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Theorem eqglact 18114
Description: A left coset can be expressed as the image of a left action. (Contributed by Mario Carneiro, 20-Sep-2015.)
Hypotheses
Ref Expression
eqger.x 𝑋 = (Base‘𝐺)
eqger.r = (𝐺 ~QG 𝑌)
eqglact.3 + = (+g𝐺)
Assertion
Ref Expression
eqglact ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Distinct variable groups:   𝑥, +   𝑥,   𝑥,𝐺   𝑥,𝑋   𝑥,𝐴   𝑥,𝑌

Proof of Theorem eqglact
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 eqger.x . . . . . . 7 𝑋 = (Base‘𝐺)
2 eqid 2778 . . . . . . 7 (invg𝐺) = (invg𝐺)
3 eqglact.3 . . . . . . 7 + = (+g𝐺)
4 eqger.r . . . . . . 7 = (𝐺 ~QG 𝑌)
51, 2, 3, 4eqgval 18112 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
6 3anass 1076 . . . . . 6 ((𝐴𝑋𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌) ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
75, 6syl6bb 279 . . . . 5 ((𝐺 ∈ Grp ∧ 𝑌𝑋) → (𝐴 𝑥 ↔ (𝐴𝑋 ∧ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌))))
87baibd 532 . . . 4 (((𝐺 ∈ Grp ∧ 𝑌𝑋) ∧ 𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
983impa 1090 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝐴 𝑥 ↔ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)))
109abbidv 2843 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → {𝑥𝐴 𝑥} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
11 dfec2 8092 . . 3 (𝐴𝑋 → [𝐴] = {𝑥𝐴 𝑥})
12113ad2ant3 1115 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = {𝑥𝐴 𝑥})
13 eqid 2778 . . . . . . . . 9 (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥))) = (𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))
1413, 1, 3, 2grplactcnv 17989 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴))))
1514simprd 488 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)))
1613, 1grplactfval 17987 . . . . . . . . 9 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1716adantl 474 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
1817cnveqd 5596 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘𝐴) = (𝑥𝑋 ↦ (𝐴 + 𝑥)))
191, 2grpinvcl 17938 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((invg𝐺)‘𝐴) ∈ 𝑋)
2013, 1grplactfval 17987 . . . . . . . 8 (((invg𝐺)‘𝐴) ∈ 𝑋 → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2119, 20syl 17 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑥𝑋 ↦ (𝑔 + 𝑥)))‘((invg𝐺)‘𝐴)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2215, 18, 213eqtr3d 2822 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2322cnveqd 5596 . . . . 5 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
24233adant2 1111 . . . 4 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → (𝑥𝑋 ↦ (𝐴 + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)))
2524imaeq1d 5769 . . 3 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌))
26 imacnvcnv 5902 . . 3 ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌)
27 eqid 2778 . . . . 5 (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) = (𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥))
2827mptpreima 5931 . . . 4 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌}
29 df-rab 3097 . . . 4 {𝑥𝑋 ∣ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌} = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3028, 29eqtri 2802 . . 3 ((𝑥𝑋 ↦ (((invg𝐺)‘𝐴) + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)}
3125, 26, 303eqtr3g 2837 . 2 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌) = {𝑥 ∣ (𝑥𝑋 ∧ (((invg𝐺)‘𝐴) + 𝑥) ∈ 𝑌)})
3210, 12, 313eqtr4d 2824 1 ((𝐺 ∈ Grp ∧ 𝑌𝑋𝐴𝑋) → [𝐴] = ((𝑥𝑋 ↦ (𝐴 + 𝑥)) “ 𝑌))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387  w3a 1068   = wceq 1507  wcel 2050  {cab 2758  {crab 3092  wss 3829   class class class wbr 4929  cmpt 5008  ccnv 5406  cima 5410  1-1-ontowf1o 6187  cfv 6188  (class class class)co 6976  [cec 8087  Basecbs 16339  +gcplusg 16421  Grpcgrp 17891  invgcminusg 17892   ~QG cqg 18059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5049  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3682  df-csb 3787  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-iun 4794  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-f1 6193  df-fo 6194  df-f1o 6195  df-fv 6196  df-riota 6937  df-ov 6979  df-oprab 6980  df-mpo 6981  df-ec 8091  df-0g 16571  df-mgm 17710  df-sgrp 17752  df-mnd 17763  df-grp 17894  df-minusg 17895  df-eqg 18062
This theorem is referenced by:  eqgen  18116  cldsubg  22422  tgpconncompeqg  22423  snclseqg  22427
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